It is easy to see that this family includes various known weighted means including the \(e\) -mixture and \(m\) -mixture for \(\alpha =\pm 1\)  in the literature of information geometry [1]}:
\((\alpha = 1)\ \ &\ m_f^{(\lambda ,1)}(a,b) = \exp \lbrace (1-\lambda )\ln a + \lambda \ln b\rbrace  \nonumber \\(\alpha = -1)\ \ &\ m_f^{(\lambda , -1)}(a,b) = (1-\lambda )a + \lambda b \nonumber \\(\alpha = 0)\ \ &\ m_f^{(\lambda , 0)}(a,b) = \Big ((1-\lambda )\sqrt{a} + \lambda \sqrt{b}\Big )^2 \nonumber \\(\alpha = 3)\ \ &\ m_f^{(\lambda , 3)}(a,b) = \frac{1}{(1-\lambda )\frac{1}{a} + \lambda \frac{1}{b}} \nonumber \\(\alpha =\infty )\ \ &\ m_f^{(\lambda , \infty )}(a,b) = \min \lbrace a, b\rbrace  \nonumber \\(\alpha =-\infty )\ \ &\ m_f^{(\lambda , -\infty )}(a,b) = \max \lbrace a, b\rbrace  \nonumber \) 
